Bridge to Abstract Mathematics: Mathematical Proof and Structures

2.4 ANALYSIS OF ARGUMENTS FOR LOGICAL VALIDITY, PART 1 (OPTIONAL) 79

question now shifts to whether we can derive s from this expanded list.

Well, from p and p -+ q, we get q. From q and q -, r, we get r. Now we

are left with r and s r. At this point, the chain of reasoning grinds to

a halt. We have no means of concluding s from r A (s + r) (check if you

wish that [r A (s -+ r)] -+ s is not a tautology). Hence we are led to

doubt the validity of the argument. To prove that nonvalidity, we must

come up with a combination of truth values for which the conditional

in question is false. This we can also do without resorting to a complete

table.

We reason as follows: We wish to find truth values for p, q, r, and s

such that the conjunction (p + q) A (q -+ r) A (s -+ r) of the partial

premises is true while the conclusion p -, s is false. Clearly p must be

true and s must be false in order for p -+ s to be false. In that case if

p -+ q is to be true, then q must be true. But then r must be true in order

for q -+ r to be true. Note that if s is false and r is true, the final partial

premise s -+ r is true. Hence we have found a combination of truth

values, namely, TTTF for p, q, r, and s, respectively, for which the premise

of the argument is true while the conclusion is false. This proves con-

clusively the nonvalidity of the argument.

You will probably enjoy the exercises that follow much more if you

employ the approach of Examples 3 and 4, rather than the boring and

mechanical truth table approach. Also, you should use throughout these

exercises the "strict mathematical" translation into symbols, given in

Remark 1, Article 2.3, for such expressions as "unless," "only if," and

"necessary."

Exercises

Analyze these arguments for logical validity:

1. Good weather is necessary for a successful garden. The garden is successful.
   Therefore the weather was good.


2. If today is Monday, then tomorrow is Tuesday. But today is not Monday. There-
   fore tomorrow is not Tuesday.


3. Either today is Monday or today is Tuesday. But today is not Monday. There-
   fore today is Tuesday.

*4. I will lose my job unless Smith is retained. He will be fired only if you recommend
it. Therefore I will keep my job if you do not recommend his firing.
5.1f5+7=12,then6>8. If5+7=7+5,then5+7=12. But5+7=7+5.
Therefore we may conclude 6 > 8. (Hint: Recall the discussion immediately follow-
ing Definition 1.)

6. If the dollar is strong, then exports decrease. Unemployment will rise unless the